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2D Gabour Filter :
In image processing a Gabor filter is a linear filter used for edge detection. Frequency and orientation representations of Gabor filters are similar to those of the human visual system, and they have been found to be particularly appropriate for texture representation and discrimination. In the spatial domain, a 2D Gabor filter is a Gaussian kernel function modulated by a sinusoidal plane ave. Formulae: Its impulse response is defined by a sinusoidal ave !a plane ave for 2D Gabor filters" multiplied by a Gaussian function. #ecause of the multiplication$convolution property !%onvolution theorem", the Fourier transform of a Gabor filter&s impulse response is the convolution of the Fourier transform of the harmonic function and the Fourier transform of the Gaussian function. 'he filter has a real and an imaginary component representing orthogonal directions. 'he t o components may be formed into a complex number or used individually. %omplex

(eal

Imaginary

here

and

In this equation,

represents the avelength of the sinusoidal factor,

represents the orientation of the

normal to the parallel stripes of a Gabor function, is the phase offset, is the sigma of the Gaussian envelope and is the spatial aspect ratio, and specifies the ellipticity of the support of the Gabor function.

Laplacian of Gaussian: Laplacian filters are derivative filters used to find areas of rapid change (edges) in images. Since derivative filters are very sensitive to noise, it is common to smooth the image (e.g., using a Gaussian filter) before applying the Laplacian. This two-step process is call the Laplacian of Gaussian (LoG) operation.

There are different ways to find an appro imate discrete convolution !ernal that appro imates the effect of the Laplacian. " possible !ernel is

This is called a negative Laplacian because the central pea! is negative. #t is $ust as appropriate to reverse the signs of the elements, using -%s and a &', to get a positive Laplacian. #t doesn(t matter. To include a smoothing Gaussian filter, combine the Laplacian and Gaussian functions to obtain a single e)uation*

Edge Detection: Edge detection algorithms operate on the premise that each pi el in a grayscale digital image has a first derivative, with regard to the change in intensity at that point, if a significant change occurs at a given pi el in the image, then a white pi el is placed in the binary image, otherwise, a blac! pi el is placed there instead. #n general, the gradient is calculated for each pi el that gives the degree of change at that point in the image. The )uestion basically amounts to how much change in the intensity should be re)uired in order to constitute an edge feature in the binary image. " threshold value, T, is often used to classify edge points. Some edge finding techni)ues calculate the second derivative to more accurately find points that correspond to a local ma imum or minimum in the first derivative. This techni)ue is often referred to as a Zero Crossing because local ma ima and minima are the places where the second derivative e)ual Fero, and its left and right neighbors are non-Fero with opposite signs. 1. Canny Edge Detector:

Finds edges by looking for local maxima of the gradient of f!x,y". 'he gradient is calculated using the derivative of a Gaussian filter. 'he method uses t o thresholds to detect strong and eak edges, and includes the eak edges in the output only if they are connnected to strong edges. 'herefore, this method is more likely to detect true eak edges.

Split-and-merge algorithm: 'he purpose of the algorithm is, given a curve composed of line segments, to find a similar curve ith fe er points. 'he algorithm defines &dissimilar& based on the maximum distance bet een the original curve and the simplified curve. 'he simplified curve consists of a subset of the points that defined the original curve.

This algorithm aims at minimiFing an objective function, in this case a s)uared error function. The ob$ective function

,

where is a chosen distance measure between a data point and the cluster centre indicator of the distance of the n data points from their respective cluster centres.
%Fet-s ma4e some fa4e data .ith t.o groups n=P); %sample si5e x=#randn(n,!)'+;randn(n,!)'2(P)$; y=#randn(n,!)'+;randn(n,!)'2(P)$;